Here the integral $\int_{-\infty}^{\infty} e^{-ac-c^2/2} dc$ can be solved with help of the solution $\int_{-\infty}^{\infty} e^{-Ax^2} e^{-2Bx}\,dx=\sqrt{\frac{\pi}{A}}e^{\frac{B^2}{A}} \quad (A>0)$ from the list of integrals of exponential functions.